Tuesday, July 03, 2007

The Area of a Circle

Triss uses Scott Foresman's Exploring Mathematics currently, and is working like fury to get through the 6th grade book before September so she can begin the University of Chicago School Mathematics Project's Transition Math text (also put out by Scott Foresman).

Today the lesson was an introduction to finding the area of a circle, and I really appreciate the way the authors laid out the explanation of the formula. I do not remember ever coming in contact with this kind of explanation or reasoning for formulas in my own math books when I was in junior high and high school. (I am of the private opinion that I might have been a math person had I had the right math teachers in school-- there is so much about math that pleases me when I finally get concepts. I coulda been a contendah! But I digress.)

I want to share their explanation:

First off, Ivey had to make a circle with a radius of four inches out of construction paper. Then she drew eight diameters through it at regular intervals. After that she cut it into "pie slices," for lack of a better description, and ended up with sixteen slices of circle. Then she was instructed to lay the slices out in a row, alternating top to bottom, so that they made a rough parallelogram.

Now here is the cool part.

They had her measure the base of the "parallelogram" and compare it to the circumference of the circle (at this point she wanted my help, and we got roughly half the circumference). Then they had us measure the height of the "parallelogram" and compare it to the radius of the circle. They were equal. Then they led us through the reasoning process that begins with the formula for the area of parallelograms, and ends with the formula for a circle. I really got excited at this point because I had never realized those two formulas connected. Here is what it looked like (I'm writing it in words below each equation to help non-math folks-- me-- understand just how cool this is):

Area = base x height
(The area is equal to the base plus times the height)

Area = (1/2 x C) x r
(Since the base and height of a circle is very difficult to measure, we have to take the circle apart and make it into something easier, ie., a parallelogram. And look! the base of the resulting parallelogram is equal to half the circumference, while the height is equal to the radius. This means that within the equation for area, we can substitute half the circumference for the base and the radius for the height. Then we won't half to continue cutting circles into pie slices and fitting them into paralellograms each time.)

A = 1/2 x (2 x pi x r) x r
(But that is a cumbersome equation because we still have to figure out the Circumference before figuring out the area. So let's take the equation for circumference, which is 2 x pi x r, combine it with the rest of the equation, and simplify it.)

A = pi x r(squared)
(Since 1/2 x 2 equals 1, and 1 times anything is equal to the anything, we can cancel out the 1/2 and the 2. Then we have pi x r x r. r x r is the same as saying "radius squared", so let's just say that. Voila!)


Pi times radius squared is the formula for circles I know from school, but I never learned how "they" decided it was the magic formula that would always result in the area of a circle. To have a math book appreciate the innate intelligence of regular students enough to explain where a formula came from is inspiring. It helps a person begin to realize that perhaps math is accessible after all, though it is also challenging.

(And yes, I know I am a geek. But I like it.)

Monday, July 02, 2007

A Mind At A Time, Ch. 3A

Chapter Three is called "Conducting a Mind" and is about attention. Dr. Levine has a phrase for someone with an attention dysfunction. He says they have inadequate brain leadership.

Attention controls help students to focus on the task or thought at hand rather than whatever flits across their minds. With weak attention controls, their thoughts are blown about as by the wind, and head in first one direction, then another, at the mercy of circumstance.

Some attention techniques:

1. Whisper directions to yourself as they are being given, to be sure you are listening carefully,

2. Go back over all your work to check for errors,

3. Ask yourself questions like, "What's the best way to do this?" or "Is this the best thing to do now?"


One thing I appreciate about Dr. Levine's discussion on attention-- he does not want to lump all children with attention struggles into one group. He uses two children as examples in this chapter, and they are different. One is a child who frequently disrupts class, and emphatically marches to the beat of his own drummer; the other is a child who is quiet and compliant, but struggles greatly with focusing her attention enough to write on a subject.

He speaks of the different processes encompassed in writing a composition:

You have to slow down, plan, organize your thinking, pace yourself, watch what you're putting on paper, and pay attention to all kinds of small details all at once (such as punctuation, spelling, capitalization, and use of grammar). (p. 55)


His recommendations for someone who struggles with writing is to do each thing in steps, with breaks between: brainstorm, write down key ideas, organize ideas into proper order, write a rough draft, and finally write a draft that is neat and spelled/punctuated correctly.

Dr. Levine points out that the performance of a child with attention troubles is inconsistent, erratic. This can be a cause of frustration for the child, parents and teachers. But children with these difficulties are often refreshingly original, interesting and extraordinary in other ways. Because of these traits, Dr. Levine theorizes that they must be wired differently, rather than having an actual dysfunction.

There are three different forms of attention control: control over mental energy, control over intake (of information, etc.), and control over output (of work/behavior).

Mental Energy Controls

A. Alertness Control: The ability to remain alert when necessary-- this can be helped by quietly tapping the child as she zones out.

B. Mental Effort Control: Apparently making an effort to do something you don't want to do is easier for some than others. I am not sure I agree with this. Doing what is right because it is right is a habit, and if it has been learned, it ought to be automatic. Admittedly, I have not studied neurology, nor medicine or psychology of any kind, other than Psych 101 in college. And in Dr. Levine's defense, he does point out that a good work ethic established at home will make mental effort easier. Sounds like habit to me.

C. Sleep-Arousal Control: Dysfunction of this control means dozing when you should be awake, and staying awake when you should be sleeping. Again, strict habits of sleeping and waking ought to be established.

D. Consistency Control: Being reliable, or not. Dr. Levine tells a story of a man who made his life accomodate his unreliability. He made money building and selling beautiful furniture, but built it on his own schedule, sleeping in when he wanted, taking days off, and then working like fury when he felt like work. It's a good thing he felt like working some, and had had the tenacity to learn his trade well before allowing himself the freedom to work or not!




The Intake Controls

A. Selection Control: This is choosing what to pay attention to. The uproarious boy Dr. Levine referred to as one of his two sample children at the beginning of the chapter described his struggle, "You know, my head is just like a TV set, but I have no remote control for it, so I get all the programs on my screen at the same time." This can occur in one of two ways: either the child pays attention to useless input such as irrelevant noise or visuals, or else the child pays attention to the input he needs, but focuses on unimportant aspects of that input. Deciding what is important and unimportant is vital. Some kids are even distracted by their own memories, or by thoughts of the future. As kids with selection control problems get older, they have a hard time determining the key points of a lesson, which can really hurt in high school and college. I think consistently practicing narration ought to help quite a bit with this. Some other things he recommends for kids with these kinds of struggles are to consciously practice taking notes, highlighting their books, assigning importance to the input they receive. One teacher had his students summarize an article in a hundred words; had them shorten it to fifty words the next week; and finally had them drop it to a twenty-five word summary, completely distilling their ideas. I remember doing this in high school English class. It was so difficult. And I am attempting something similar with this book!

B. Depth and Detail Control: Preventing input from going "in one ear and out the other." People like this tend to get the big picture, but miss out on pertinent details. They leave out substantives (names, etc.). I would guess these folks use words like "stuff" and "things" a lot. An opposite problem is allowing details to penetrate too deeply. This causes a person to think too much. Kids like this are painfully slow at getting work done, preoccupied with every little detail. It seems one must strike a balance. All of this appears very subjective to me. It really depends on what the teacher wants in a class, doesn't it? Perhaps the child is meant to think deeply on a particular subject, but the class does not delve that deeply; is the child wrong? Kids must abide by the standards of the teacher or else risk a poor grade. I know we are all thankful that Einstein thought as deeply as he did, although it did not help him in class!

C. Mind Activity Control: Connections, connections. An active mind is constantly attempting to connect what it is learning to what it already knows. A passive mind allows new input to "bob around on the surface," seldom connecting to anything else. But even active minds have to beware-- if the mind is too active, it produces a wave of connections that are only meaningful to the individual-- speaking of a princess reminds a child of Cinderella and the glass slipper, and then she remembers she got new shoes yesterday, which reminds her of the ice cream flavor she chose when they got a treat afterward-- "If You Give A Mouse A Cookie" kind of stuff. Often, kids like this have great imaginations. They just need to train themselves to take their mind journeys at times other than class time.

D. Span Control: This can be not focusing long enough, or focusing too long-- for instance, if a child has trouble transitioning from playtime to dinner, or from dinner to homework. Working with a timer, allowing the child to know how long a task will take, is very helpful for kids like this.

E. Satisfaction Control: This one makes me laugh. If only we all had better satisfaction control!! O, what a world this would be! But seriously, these are the kids who crave excitement, who find it hard to be satisfied with daily existence. Insatiability. Dr. Levine recommends that these kids be allowed to pursue their own interests outside of class, within reason.


Are you still with me? I know this is a long post. I have two more sections to summarize, but I will put those in a second post.

Sunday, July 01, 2007

I Love To See The Lord Below

The DHM has linked to a couple of articles on praise music v. hymns at the bottom of her Sunday Hymn Post this morning, and I really do appreciate those articles. We are hymn-only, no-instruments-in-church type folks, and I truly appreciate the attempts of folks outside our tradition to nail down what is irksome about praise music. It does not come across our radar enough for us to say.

Patrick O'Hannigan at The Paragraph Farmer presents a Catholic's point of view on this kind of music. When I came to his take on how praise music came to be, I knew I had to blog about it:

They simply wanted to "reach people where they're at," and figured that grand old hymns had to go, if for no other reason than that they harkened back to the days of what singer/songwriter John Prine called "stained glass in every window, [and] hearing aids in every pew."


I wish some church music ministers had been at the church meeting we attended this weekend. Although not large by mainstream Christian standards (I'm guessing around 250-300 people attended), there were folks with wheelchairs and hearing aids, yes-- but there were also couples with infants; mothers toting toddlers; schoolchildren with their parents, grandparents, cousins and friends; teenagers congregating in various pews; and even young unmarried adults-- some, college students, others young working folks.

And they were singing loud. Don't get me wrong, it wasn't shouting or poor quality noise. It was a unified swell of song that rung to the ceiling and out through the cracks in the doors. And I am not talking about a few strategically placed well-trained singers. Even the little children and old folks were getting after it. These folks are trained, all right-- in a strong congregational hymnsinging tradition that has not been altered for centuries. This is an environment where you can really get into the spirit and sing out, and you don't need to worry about dwarfing another part, or embarrassing yourself with a missed note. There was no quenching of the spirit in this congregation.

And let this feeble body fail,
And let it faint and die
My soul shall quit this mournful vale

And soar to worlds on high

Shall join the disembodied saints
And find its long-sought rest
That only bliss for which it pants
In the Redeemer's breast.

And I'll sing hallelujah!
And you'll sing hallelujah!
And we'll all sing hallelujah!
When we arrive at home!


I just cannot quantify and classify the experience of sitting in a churchhouse with hundreds of likeminded people who all know the same hymns in four-part harmony and join their hearts in worship to the true and living God. Glorious. And the sound system was only used for the preaching. If they had had mikes on the congregational singing, the roof would probably have caved in.

I have friends who think we are odd for going to so many church meetings. After all, there is no nursery, no children's church, no Veggie Tales. The kids sit with us during each service, or else sit with their friends (in a nearby pew, within thumping distance, you understand), or with other older members they have become attached to (sometimes relatives, sometimes not). We attend three church services per day at these meetings (except the first night and on Sunday), and the sermons are an hour long (longer if there are two preachers).

All I can say is come and see. There is a spiritual quality to the worship and a love in the fellowship that cannot be explained in conversation, or even a blog post. Powerful. Even the children drink it in, though they may not understand every word. We could not stay for the evening service last night, and our girls were disappointed. Despite long hours of sitting still and paying attention, they enjoy these meetings.

I love to see the Lord below
His church displays Hi Grace;
But upper worlds His glory know,
And view Him face to face.

I love to worship at His feet,
Though sin annoy me there;
But saints exalted near His feet
Have no assaults to fear.

I love to meet Him in His courts,
And taste His heavenly love;
But still His visits seem too short,
Or I too soon remove.

O Lord, I love Thy service now;
Thy church displays Thy power;
But soon in heaven I'll to Thee bow,
And praise Thee evermore.